for all integers \(i,j\ge 0\) . Piling \(\pi _{i,j}\)  cubes over position \((i, j)\)  gives a 3D Young diagram. 3D Young diagrams arose naturally in the melting crystal model[1]}, [2]}. We always identify 3D Young diagrams with plane
partitions as explained above. For example, the 3D Young diagram
\(\begin{tikzpicture}(0,0)rectangle(0.5,0.25);(0.25,0) -- (0.25,0.25) [-];(0,0.25) -- (0.175,0.35) [-];[shift = {+(0.25,0)}](0,0.25) -- (0.175,0.35) [-];[shift = {+(0.5,0)}](0,0.25) -- (0.175,0.35) [-];[shift = {+(0.5,-0.250)}](0,0.25) -- (0.175,0.35) [-];(0.175,0.35) -- (0.675,0.35) [-];[shift = {+(0.25,0)}] (0.425,0.35) -- (0.425,0.1) [-];\end{tikzpicture}\) 
can also be denoted by the plane partition
\((1,1)\) .
